Linear time algorithms for NP-hard problems restricted to partial k-trees
Discrete Applied Mathematics
The traveling salesman problem with distances one and two
Mathematics of Operations Research
Beyond uniformity and independence: analysis of R-trees using the concept of fractal dimension
PODS '94 Proceedings of the thirteenth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems
Accessing nearby copies of replicated objects in a distributed environment
Proceedings of the ninth annual ACM symposium on Parallel algorithms and architectures
Approximation schemes for Euclidean k-medians and related problems
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Approximating geometrical graphs via “spanners” and “banyans”
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
Journal of the ACM (JACM)
Finding nearest neighbors in growth-restricted metrics
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Distributed object location in a dynamic network
Proceedings of the fourteenth annual ACM symposium on Parallel algorithms and architectures
When Hamming Meets Euclid: The Approximability of Geometric TSP and Steiner Tree
SIAM Journal on Computing
Estimating the Selectivity of Spatial Queries Using the `Correlation' Fractal Dimension
VLDB '95 Proceedings of the 21th International Conference on Very Large Data Bases
Polynomial-Time Approximation Schemes for the Euclidean Survivable Network Design Problem
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
A Nearly Linear-Time Approximation Scheme for the Euclidean kappa-median Problem
ESA '99 Proceedings of the 7th Annual European Symposium on Algorithms
Deflating the Dimensionality Curse Using Multiple Fractal Dimensions
ICDE '00 Proceedings of the 16th International Conference on Data Engineering
Clique is hard to approximate within n1-
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Probabilistic approximation of metric spaces and its algorithmic applications
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Navigating nets: simple algorithms for proximity search
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Bypassing the embedding: algorithms for low dimensional metrics
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Triangulation and Embedding Using Small Sets of Beacons
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
A tight bound on approximating arbitrary metrics by tree metrics
Journal of Computer and System Sciences - Special issue: STOC 2003
Fast construction of nets in low dimensional metrics, and their applications
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
On hierarchical routing in doubling metrics
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Distance estimation and object location via rings of neighbors
Proceedings of the twenty-fourth annual ACM symposium on Principles of distributed computing
On the locality of bounded growth
Proceedings of the twenty-fourth annual ACM symposium on Principles of distributed computing
Name independent routing for growth bounded networks
Proceedings of the seventeenth annual ACM symposium on Parallelism in algorithms and architectures
A linear-time approximation scheme for planar weighted TSP
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Searching dynamic point sets in spaces with bounded doubling dimension
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The black-box complexity of nearest-neighbor search
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
Cover trees for nearest neighbor
ICML '06 Proceedings of the 23rd international conference on Machine learning
Optimal-stretch name-independent compact routing in doubling metrics
Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing
Routing in Networks with Low Doubling Dimension
ICDCS '06 Proceedings of the 26th IEEE International Conference on Distributed Computing Systems
On sampling in higher-dimensional peer-to-peer systems
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
An approximation algorithm for a bottleneck traveling salesman problem
Journal of Discrete Algorithms
Proximity algorithms for nearly-doubling spaces
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
The traveling salesman problem: low-dimensionality implies a polynomial time approximation scheme
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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The Traveling Salesman Problem (TSP) is a canonical NP-complete problem which is known to be MAX-SNP hard even on (high-dimensional) Euclidean metrics [39]. In order to circumvent this hardness, researchers have been developing approximation schemes for low-dimensional metrics [4, 38] (under different notions of dimension). However, a feature of most current notions of metric dimension is that they are "local": the definitions require every local neighborhood to be well-behaved. In this paper, we consider the case when the metric is less restricted: it has a few "dense" regions, but is "well-behaved on the average"? To this end, we define a global notion of dimension which we call the correlation dimension (denoted by dimC), which generalizes the popular notion of doubling dimension. In fact, the class of metrics with dimC = O(1) not only contains all doubling metrics, but also contains some metrics containing uniform submetrics of size √n. We first show, using a somewhat "local" argument, that one can solve TSP on these metrics in time 2O(√n); we then take advantage of the global nature of TSP (and the global nature of our definition) to give a (1 + ε)-approximation algorithm that runs in sub-exponential time: i.e., in 2O(nδε-4dimC)-time for every constant 0